Consider a typical self-powered IoT node that consists of an IoT sensor node, an energy harvesting system, and an ESS, as shown in Figure 1. Energy is harvested from ambient or external sources (e.g., solar, artificial light, radio frequency, and vibration) to power the sensor node directly. Any residual energy is stored in an ESS. The stored energy is used to power the sensor node when the energy harvester is not able to generate enough energy to meet the energy needs of the node due to unfavourable environmental conditions (e.g., during the night in the case of solar energy harvesters). When the sensor node is not performing sensing, computing, or processing operations, it is forced into the sleep mode, where it consumes negligible amounts of energy. Figure 2 shows a snapshot of the power profile of an IoT node consisting of two modes: sleep mode (when it is not performing sensing, computing, or communication functions) and active mode (when it wakes up to perform sensing, computing, or communication operations). From the power profile, the average power consumption of the node is given in Equation 1: P node = D ⋅ P act + 1 − D ⋅ P sleep , (1) where the duty cycle ratio is given in Equation 2, D = t act t act + t sleep . (2)

where t act is the time spent in the active mode and t sleep is the time spent in the sleep mode. P act is the power consumption of the node in the active mode and P sleep is the power consumption of the node in the sleep mode. The energy consumed during the active mode is the sum of the energy consumed by the sensing, computing, communication units, and other auxiliary electronics components of the node during the active period.

The power profile in Figure 2 illustrates the characteristics of the IoT energy consumption model, which forms the basis of our energy packetization or quantization model in the following subsection. The power profile is obtained using a laboratory testbed that consists of two IoT nodes positioned 2 m apart along a high-pressure plastic pipe measuring 12 m in length and with a diameter of 25 mm. In order to optimize or minimize the energy consumption of the IoT nodes, the nodes are configured to perform distributed computing with Kalman filtering (by sharing the computing load), adaptive sensing (by using an energy-efficient but less accurate accelerometer sensor and an energy-hungry but more accurate accelerometer sensor), and duty cycling (forcing the node to enter sleep modes when it is idle).

Performing energy planning of self-powered IoT nodes requires an estimate of the energy demand, energy generation, and storage capacity to ensure a low probability of service outage and a long lifetime for the node. From the characterization of the energy harvesting system (e.g., solar cells, piezoelectric, RF, or thermoelectric energy harvester), the power profile can be obtained. An empirical power profile of a solar energy harvester for an IoT node is shown in the study by Kuzman et al. (2019), which consists of active periods of solar power generation (when there is enough solar radiation) and a period of no solar power generation (when there is insufficient solar radiation, notably during the night). From the energy consumption and generation profile, the mean energy produced and consumed can be estimated. The mean energy generated and consumed can be used to determine the number of energy packets produced and consumed per unit of time, as discussed in the next section.

In order to discretize or quantize energy into energy packets, the first step is to determine the quantization step, which, in our case, is the size of the energy packet. We consider an energy packet (in mWh or mAh) as a pulse of power or current which lasts for a defined time duration. Assuming that energy is consumed during active periods when the node wakes up to perform sensing, computing, or communication (and that a negligible amount of energy is consumed during the deep sleep period), the size of the energy packet can be considered to be E p = P act ⋅ t act . However, the quantization step can be set to any arbitrary value but must be kept consistent in the quantization of the energy harvesting, consumption, and storage processes, as in the study by Da Silva et al. (2017).

Let C B (measured in mWh) represent the capacity of the ESS, which may be a battery or a supercapacitor. Then, the capacity of the ESS (in energy packets) is B = C B / E p ; that is, the number of energy packets that can be stored in the ESS is B, and the energy states of the ESS are { 0,1,2 , … , B } . We assume that the node wakes up only when triggered by a random event (e.g., leakage of fluids from a pipe in the case of a pipeline monitoring system). In this case, the energy drawn from the battery per time unit is scattered independently and uniformly in the sense of a Poison process (Kaj and Konané, 2016). The energy consumption process becomes E node = t act ⋅ P act N t ( 1 / t i ) , where N t μ denotes a standard Poisson process on the half line with constant intensity μ . That is, energy is drawn from the battery in small jumps of energy E p = t act ⋅ P act , which occur interspaced by independent and exponentially distributed waiting times with expected value t i = t act + t sleep . From the power consumption profile, the mean number of energy packets drawn from the ESS per time unit in the time interval [ 0 ,   t i ] is μ = t act t act + t sleep ⋅ P act E p . (3)

We consider an intermittent energy harvesting source (e.g., the presence of solar radiation, light, vibration, wind, RF radiation, and heat). For simplicity, we assume that the energy arrival times of the energy packets follow a Poisson process with rate λ H (Ng et al., 2013; Wang et al., 2014). This assumption may be realistic in self-powered IoT nodes that stay in a deep sleep mode for a time that is exponentially distributed. They wake up to receive or transmit data packets, harvest wireless RF energy at the same time, and then return to the sleep mode. From the power generation profile of the energy harvester, the number of energy packets generated per time unit in the time interval [ 0 ,   T ] is as follows: λ H = 1 E p ⋅ T ∫ 0 T P H τ   d τ . (4) Here, P H ( t ) is the output energy profile of the energy harvesting system. If the harvested energy is greater than the energy required to power the IoT node, the surplus is stored in the battery to be used when the node’s needs are greater than the energy production. From the energy conservation principle and assuming that there is no energy leakage from the ESS, the mean number of energy packets delivered to the battery is λ = μ − λ H , which also follows a Poisson process. In this case, the process of delivering energy packets of the battery is also assumed to follow a Poisson process and from Equations 3, 4, the mean rate of delivering energy packets to the battery is λ = λ H − μ .

The ESS storage space is partitioned into m non-overlapping intervals called energy-saving regimes by introducing m − 1 energy-saving thresholds (or barriers or switches). In the m t h interval (with the highest energy content), the IoT node is fully functional and performs all its functions typically. However, in the subsequent intervals, some of the functionalities of the node may be limited or disabled to save energy to prolong the lifetime of the device, making the node semi-functional. In the first interval (with the lowest energy content), most of the functionalities (computation and communication) of the nodes are significantly limited or disabled; that is, the node is non-functional. Therefore, the node’s mean rate of energy consumption depends on the energy content of the energy storage system as shown in Equation 5: μ n = μ 1          0 < n ≤ K 1 , μ 2          K 1 < n ≤ K 2 , μ 3          K 2 < n ≤ K 3 , ⋯          ⋯   , μ m          K m < n ≤ B . (5)

By introducing energy thresholds and reducing energy consumption at the node as the energy content of the ESS goes below the various thresholds, the lifetime of the node can be increased. For specific IoT sensors, energy consumption can be reduced on the fly by throttling the speed of the processor clock, decreasing the operating voltage, or decreasing the transmission power. The drawback of forcing the node to enter into energy-saving modes is that it may degrade the quality of service of the nodes. This should only be considered when the energy stored in the ESS is below certain critical thresholds, and sacrificing some level of performance is acceptable. Energy modes for some IoT devices may include the following: run mode (CPU, flash, SRAM, and peripheral on), doze mode (CPU clock runs slower than peripheral on), idle mode (CPU off, flash, SRAM, and peripheral on), sleep mode (CPU, flash, SRAM off, and peripheral on), and deep sleep mode (CPU, flash, SRAM, and peripheral off) (Evanchuk, 2024).

In the ESS model, we assume that energy is delivered and consumed by quantum (energy packets). The process resembles the behaviour of a queueing system. The energy packets are like customers, and the time it takes to consume one packet corresponds to the service time. The number of customers in the queueing system denotes the energy in the ESS. It allows us to make use of the existing queueing models. We model the dynamic changes in the number of energy packets in the ESS as an M/M(n)/1/B queueing Markov process { N ( t ) | t ≥ 0 } such that p ( n , t ) = Pr { N ( t ) = n } is the probability of having n energy packets in the ESS. In the notation, based on the study by Kendall )53), it is a station with Poisson input, exponentially distributed service time, single server, and limited to B number of customers inside. M(n) underlines that the parameter μ of the time to consume an energy packet may depend on the queue length (number of energy packets in the ESS), μ = μ ( n ) . The model consists of a set of equations (see, e.g., Kleinrock (1975)), as follows: d p 0 , t d t = − λ p 0 , t + μ 1 p 1 , t , d p n , t d t = − λ + μ 1 p n , t + λ p n − 1 , t + μ n p n + 1 , t , n = 1 , … B − 1 , d p B , t d t = λ p B − 1 , t − μ B p B , t . (6)

This system has a well-known solution, both in transient and steady states, if the parameter μ does not depend on n , but in the case of μ ( n ) , the solution is limited to a steady state when state probabilities do not depend on time. Therefore, in Section 3.2, we analyze its transient state in detail. The model can be extended to the case where the distributions between the time of arrival of energy packets and the distributions of the time of their consumption are not exponential but are a linear combination of exponentially distributed phases that can approximate any distribution. Many software tools adapt the parameters of such distributions to the actual measurement data (e.g., Asmussen et al. (1990) and Bause et al. (2010)), as well as tools to numerically solve the resulting Markov chain equations (e.g., Prism, Kwiatkowska et al. (2011), or our Olymp, Pecka et al. (2018)).

In the steady state, when lim x → ∞ p ( n , t ; n 0 ) = p ( n ) , the differential equations above become linear equations which can be easily solved to derive the steady-state distribution of the number of energy packets in the ESS and the probability p ( 0 ) of depleting all the energy packets stored in the ESS (probability that the ESS is empty). The steady-state distribution of the number of energy packets in the ESS is as follows (e.g., Kleinrock, 1975): p n = p 0   λ n μ 1   ⋯   μ n , (7)

and taking normalization ∑ n = 0 B p ( n ) = 1 , the probability p ( 0 ) of depleting all the energy packets stored in the ESS is as follows: p 0 = 1 1 + ∑ n = 1 B λ n / ∏ i = 0 n − 1 μ i + 1 .

From the Equation 7 above, the steady-state probability p ( B ) that the ESS is full can be derived. The energy storage space of the ESSs (e.g., battery or supercapacitor) for IoT nodes is very limited (especially for very small and mobile IoT nodes), and energy packets that arrive when the ESS is full are lost, resulting in undesirable energy wastage. In addition, when all the energy packets stored in the ESS are depleted, the node shuts down, interrupting the service provided by the node. Thus, the probability p ( 0 ) is a critical performance metric and can be considered the service outage probability. In the case of a single threshold, K , there are two energy consumption regimes with μ ( u ) = μ 1 (for n < K ) and μ ( n ) = μ 2 (for n > K ), and the performance metrics are also a function of the energy threshold K.

We present the transient-state analysis of the energy performance of the ESS with energy thresholds. The steady-state analysis assumes that the mean rate at which energy packets are delivered to the ESS and the mean rate at which energy packets are consumed from the ESS are constant. However, the mean number of energy packets harvested may vary within a 24-h day period and between various days and months. In the case of solar energy harvesters, sufficient energy is generated during the solar hour period of the day, and no energy is generated at night. There are also fluctuations within the day that may result in fluctuations in the mean number of energy packets harvested and the mean number of energy packets delivered to the ESS. These time-dependent changes in the number of energy packets harvested and delivered to the ESS make transient analysis of the dynamic changes in the energy content of the ESS interesting. In the transient-state analysis, the performance metrics considered in the previous section in the steady-state analysis become time-dependent.

Transient analysis of M/M/1/B was performed in the study by Tákacs (1962), Morse (1958), Sharma and Gupta (1982), and recently in the study by Massey et al. (2023). Here, we extend it to the case of M/M/(n)/1/B, that is, state-dependent parameters μ ( n ) . The most straightforward approach is to consider Equation 6 in Laplace domain as shown in the set of equations in Equation 8. s P 0 , s − p 0,0 = 1 − λ P 0 , s + μ 1 P 1 , s s P n , s − p n , 0 = − λ + μ n P n , s + λ P n − 1 , s + μ 1 P n + 1 , s    1 ≤ n < B s P B , s − p B , 0 = λ P B − 1 , s − μ B P B , s    n = B . (8) Here, P n , s = L p n , t = ∫ 0 ∞ e − s t p n , t d t  and  L p n , t d t = s P n , s − p n , 0 ,

Solve the system Equation 8 for the values of s needed by the inversion algorithm, and then the originals of P ( n , s ) are looked for numerically, for example, with the use of the Stehfest algorithm (Stehfest, 1970): p n , t = ln ⁡ 2 t ∑ i = 1 N V i P n , s = ln ⁡ 2 t i ,

and V i = − 1 N / 2 + i ∑ k = i + 1 2 min i , N / 2 k N / 2 + 1 2 k ! N / 2 − k ! k ! k − 1 ! i − k ! 2 k − i ! .

In our numerical computations, we used N = 20 .

However, we also present the explicit expressions for P ( n , s ) . Below, we do it for the case when the buffer is initially empty, p ( 0,0 ) = 1 . Similarly, results can be obtained for a ‘mirror’ process that starts at B and ends at 0.

We assume that μ ( n ) takes m values specific for m zones, as defined in Equation 5. Starting from the equations in the first interval (e.g., 1 ≤ n ≤ K 1 − 1 ), λ P n − 1 , s P n , s = s + λ + μ 1 − μ 1 P n + 1 , s P n , s . (9)

Dividing both sides of Equation 9 by μ 1 , we get the following: λ μ 1 P n − 1 , s P n , s = s μ 1 + λ μ 1 + 1 − P n + 1 , s P n , s . (10)

From Equation 9, P n + 1 , s P n , s = λ s + λ + μ 1 − μ 1 P n + 2 , s P n + 1 , s . (11)

Substituting (Equation 11) in (Equation 10), we get Equation 12: λ μ 1 P n − 1 , s P n , s = s μ 1 + λ μ 1 + 1 − λ s + λ + μ 1 − μ 1 P n + 2 , s P n + 1 , s , (12) which can be rearranged as follows: λ μ 1 P n − 1 , s P n , s = s μ 1 + λ μ 1 + 1 − λ μ 1 s μ 1 + λ μ 1 + 1 − P n + 2 , s P n + 1 , s . (13)

The ratio λ μ 1 P ( n − 1 , s ) P ( n , s ) in Equation 13 can be expressed as a hypergeometric series as follows: λ μ 1 P n − 1 , s P n , s = s μ 1 + λ μ 1 + 1 − λ μ 1 s μ 1 + λ μ 1 + 1 − λ μ 1 s μ 1 + λ μ 1 + 1 − ⋯ . (14)

We apply the concepts of hypergeometric functions (Lorentzen and Waadeland, 1992) and finite continued fractions (Waadeland and Lorentzen, 2008; Ikenaga, 2022, accessed on 12 February, 2022) to simplify the hypergeometric series in Equation 14. Let x = s μ 1 + λ μ 1 + 1 − λ μ 1 s μ 1 + λ μ 1 + 1 − λ μ 1 s μ 1 + λ μ 1 + 1 − ⋯ ,

which can also be expressed as follows: x = a + b − b a + b − b a + b − b a + b − ⋯ , where a = s μ 1 + 1 and b = λ μ 1 . As x contains a copy of itself as the bottom of the first fraction, it can be expressed as follows: x = a + b − b x . (15)

The roots of Equation 15 are as follows: x = a + b ± a + b 2 − 4 b 2 .

As the fraction is positive, we take the positive root of Equation 16, x = a + b + a + b 2 − 4 b 2 . (16)

From Equations 14, 16 P n , s = 2 b a + b + a + b 2 − 4 b P n − 1 , s . (17)

Therefore, for 1 ≤ n < K 1 , the transient state probabilities P ( n , s ) are derived from Equation 17 P n , s = λ μ 1 1 x n , (18) where x = s + λ + μ 1 + s + λ + μ 1 2 − 4 λ μ 1 2 μ 1 .

Applying the above solution in Equation 18 iteratively for all intervals, we obtain the transient state probabilities are given in Equations 19–26: P n , s = λ μ 1 1 α 1 s n P 0 , s ,    1 ≤ n ≤ K 1 , λ μ 2 1 α 2 s n P 0 , s ,    K 1 < n ≤ K 2 , λ μ 3 1 α 3 s n P 0 , s ,    K 2 < n ≤ K 3 , ⋯ ⋯   , λ μ m − 1 1 α m − 1 s n P 0 , s ,    K m − 2 < n ≤ K m − 1 , λ μ m 1 α m s n P 0 , s ,    K m − 1 < n ≤ B − 1 . (19) where α i s = s + λ + μ i + s + λ + μ i 2 − 4 λ μ i 2 μ i ,     i = 1,2,3 , ⋯ m .

From first Equation 8, s + λ P 0 , s = 1 + μ 1 P 1 , s ,

we obtain P ( 0 , s ) . P 0 , s = a + b + a + b 2 − 4 b s + λ a + b + a + b 2 − 4 b − 2 λ , (20)

Equation 20 can be rearranged to obtain the transient probability of depleting all the energy packets stored in the ESS given in Equation 21: P 0 , s = s + λ + μ 1 + s + λ + μ 1 2 − 4 λ μ 1 s + λ s + λ + μ 1 + s + λ + μ 1 2 − 4 λ μ 1 − 2 λ μ 1 . (21)

From last Equation 8, s + μ m P B , s = λ P B − 1 , s ,

and the transient proability that the ESS is charged to its full capacity (and additional incoming energy packets are lost), P ( B , S ) is given in Equation 22: P B , s = λ s + μ m λ μ m 1 α m s B − 1 P 0 , s . (22)

We remind that in the case of an M/M/1/B model, the transient solutions obtained in Sharma and Gupta (1982) for the same initial condition p ( 0,0 ) = 1 and p ( n , 0 ) = 0 , n = 1 , … B is present in is present in Equation 23 P n , s = α β n α B − n + 1 − β B − n + 1 − α β n + 1 α B − n − β B − n s α B + 1 − β B + 1 , (23) where α s = s + λ + μ + s + λ + μ 2 − 4 λ μ 2 μ ,

and β s = s + λ + μ − s + λ + μ 2 − 4 λ μ 2 μ .

Similarly, P 0 , s = α B + 1 − β B + 1 − α β α B − β B s α B + 1 − β B + 1 , (24)

and P B , s = α β B α − β s α B + 1 − β B + 1 . (25)

For very large values of B, the transient solution reduces to an M/M/1 model as follows: lim B → ∞ P n , s = 1 − β ϱ n s α n , (26) where ϱ = λ / μ is the energy supply to demand ratio.

For other initial conditions, the system of Equation 8 is solved numerically.

The mean number of energy packets in the ESS at time t is E N t = ∑ n = 0 B n p n . t . (27)

The Laplace transforms above can be inverted numerically using the Stehfest algorithm to obtain p ( n , t ) , from which time-dependent performance metrics such as the service outage probability p ( 0 , t ) , energy wastage probability p ( B , t ) , and the mean number of energy packets in the ESS E [ N ( t ) ] can be obtained.

We investigate the impact of the threshold energy management policy on the device’s lifetime. The objective of introducing the adaptive threshold (or imposing the energy-saving regimes) is to increase the device’s lifetime. The device’s lifetime is the time required to deplete all the energy packets stored in the ESS (Kuaban et al., 2023b; Czachórski et al., 2022). We model the device’s lifetime as the first passage time of the M/M(n)/1/B model from any starting state to n = 0 . The density of the first passage time, γ i , 0 ( t ) , of the process that starts at n = i and is absorbed at n = 0 can be obtained numerically by solving the proposed M/M(n)/1/B model.

We compute the first passage time from B to zero of the proposed M/M/(n)/1/B model by making state zero the absorbing one, that is, modifying the first equation of the system (6) to the following form: d p 0 , t d t = μ 1 p 1 , t .

If the p ( 1 , t ) is computed for the chain initiated from state B , the intensity of entering state 0 in the equation above is the density of the first passage time from B to 0, γ B , 0 t = μ 1 p 1 , t . (28)

Similarly, to model the first passage time from 0 to B (the time required to charge the ESS to its full capacity), we make B the absorbing state and compute p ( B − 1 , t ) λ in the chain initiated at state 0. γ 0 , B t = λ p B − 1 , t . (29)

The performance metrics γ B , 0 (lifetime of the node) and γ 0 , B expressed in Equation 28 and 0; B expressed in Equation 29 can be obtained numerically using a Markov solver developed in Pecka et al. (2018).

The steady-state and transient-state analyses presented in Section 3 above are more applicable to non-solar energy sources. That is, energy sources that can produce energy both in the day and in the night (e.g., RF, vibration, and wind). Figures 3–6 present the results obtained using the analytical models presented in the previous section.

Figure 3 presents the changes of the service outage probability, p ( 0 , t ) , as a function of time until attaining a steady state. The results compare two cases: one with a threshold and the other without a threshold. The introduction of a threshold significantly reduces the probability of service outage, p ( 0 , t ) .

Figure 4 illustrates the above solution in the case where the battery volume is B = 100 energy units. The only threshold is placed at K = 40 , the consumption rates are μ ( n ) = μ 1 = 3 energy units per time unit, n = 1 , … K ; if μ ( n ) = μ 2 = 5 energy units per time unit, n = 41 … B . Depending on the value of harvesting rate λ , the probability mass of p ( n ) is concentrating close to 0 ( λ < μ 1 ) , close to B ( λ > μ 2 ) , or around K ( μ 1 < λ < μ 2 ) . If λ = μ 1 . The distribution does not change in the corresponding interval or λ = μ 2 .

Figures 5, 6 display the density of the lifetime of the IoT node γ B , 0 ( t ) for various values of the threshold K = 20,40,60 and various values of μ 1 = 2.5 , 3,4 ; μ 2 = 5 is not changing. The densities are compared with the same density when there is no threshold, and the unique rate is μ = 5 . The impact of the energy saving of the threshold K and reduced consumption rate μ 1 is essential. It influences not only the mean time to depletion but also the variance of the distribution.

The energy produced by non-solar energy sources is relatively small and may be insufficient for some energy-hungry IoT nodes. A scalable approach to generate sufficient energy to power an IoT node is the use of solar energy. However, solar energy sources produce energy during the day and not during the night; that is, when energy is equally consumed. Thus, the analysis presented in Section 3 is not sufficient to analyze ESSs that are supplied by energy from solar energy sources.

As solar energy sources generate energy during the day, part of which is used to supply the node and the residue is stored to be consumed during the night, we propose an approach that takes into consideration the day and night cycles. We assume the interleaving of day and night periods of constant duration T d and T n ; T d + T n = 24 hours. That is, the ESS is charged during the day for a duration of T d and discharged during the night for a duration of T n (e.g., T d = T n = 12 ).

In this case, the performance metrics are obtained by numerically solving differential Equation 6 considering various initial conditions. During the day, the ESS is charged with a mean rate of λ = λ H − μ ( n ) , and during night, λ = 0 and the ESS is discharged at a mean rate of μ ( n ) . The energy distribution at the end of a day makes the initial condition for the next night’s solution, and vice versa. A few figures below present the numerical solutions for several consecutive night and day periods until we observe a repetitive behaviour of the ESS, independent of its state at the very beginning. In the 24-h cycle, we assume an equal day and night duration of 12 h. It, of course, may be easily changed.

Figure 7 presents the changes of the mean number of energy packets E [ N ( t ) ] , both for the primarily empty ( n 0 = 0 ) and full ( n 0 = B ) ESS in case of two input flows λ = 8 , λ = 12 . After several periods, the influence of the initial value is not visible. The transient mean number of energy packets in the ESS is obtained using Equation 27.

Figure 8 refers to a case when the input rate is insufficient to maintain the performance of the node: in two cycles, the content of the initially full ESS goes down so that charging during the day is not sufficient to ensure continuous operation of the device. Two scenarios are considered: without threshold ( K = 0 ) and with one threshold ( K = 40 ) . The flow is too weak to ensure good performance of the ESS even with the threshold; however, its presence increases the node’s availability (when all the stored EPs are consumed).

Figure 9 illustrates the impact of the threshold position, K = 20,40,80 , in case when the input rate is relatively high ( λ = 8 ) . The earlier the device enters into the energy efficient regime, the higher the number of EPs stored during the cycle.

Figure 10 shows the influence of input rate λ for a given threshold ( K = 40 ) on the mean energy content during nights and days. λ = 8,12 provide a safe energy reserve over the entire operating cycle, whereas lambda = 4 no longer ensures continuous operation. However, we should remember that the above curves refer to average energy values; analysis of the probability of a battery being empty or full as a function of time gives us more specific information.

In Figure 11, the probability of service outage, p ( 0 , t ) , is presented if, at the beginning, the ESS is full and the energy input rate is insufficient λ = 4 . During the second night, the probability of complete depletion mounts sharply, and putting the threshold at K = 40 does not sufficiently improve the situation.

Figures 12, 13 refer to the time-varying probability that the battery is fully charged and further energy packets delivered are lost. In the first figure, in the presence of input rate λ = 8 , we observe how lowering the threshold from position K = 80 to K = 40 reduces the probability of having a full ESS. In the second figure, we see the impact of increasing the input rate on the probability p ( B , t ) for a fixed value K = 40 of threshold. Only for the highest value λ = 12 , the probability of fully charged ESS at the end of the day is significant.

Finally, Figures 14, 15 refer to the lifetime of the IoT node showing the density γ B , 0 ( t ) of the first passage time from B to 0. The flow λ = 4 is insufficient to maintain the uninterrupted work of the ESS. With the threshold K = 40 , the working time is longer and may last several periods; we see how nights and days shape the curve of the density. The second figure displays the same curves on logarithmic scale to demonstrate that the model is able to predict very weak probabilities.